)
)
(
33
)
The corresponding value of Y is given by: Y=X?X
0+Y
0.
Varying the parameter X in Equation (32) and then inverting z=z(X) to X=X(z) can get the solutions for X(z), which also gives Y(z) and θd(z) from Equation (33). In this way, the divergence angle and the semi-axes as a function of z along the charged-particle beam can be plotted, as shown in FIGS. 18A-18B, for electron beams of kinetic energies 3.7 MeV (panel a) and 100 eV (panel b).
As can be seen from the FIGS. 18A-18B, the large Lorentz factor of the relativistic 3.7 MeV electrons permits an even larger current to be used without causing the beam to diverge significantly over the interaction distance. The divergence angle of the 100 eV electron beam remains reasonably small over the interaction region, but additional beam-focusing stages may probably be needed for larger currents or longer interaction distances.
When X-X0<<(X0+Y0)/2, as is the case in the plots of FIGS. 18A-18B, Equations (32) and (33) can be simplified via Taylor expansions to obtain analytical expressions of X, Y and θd as functions of z:
